|
%I #155 Feb 12 2024 08:50:27
%S 0,0,0,1,3,5,7,10,14,18,22,27,33,39,45,52,60,68,76,85,95,105,115,126,
%T 138,150,162,175,189,203,217,232,248,264,280,297,315,333,351,370,390,
%U 410,430,451,473,495,517,540,564,588,612,637,663,689,715,742,770,798
%N a(n) = floor(binomial(n, 2)/2).
%C Column sums of an array of the odd numbers repeatedly shifted 4 places to the right:
%C 1 3 5 7 9 11 13 15 17...
%C 1 3 5 7 9...
%C 1...
%C .........................
%C -------------------------
%C 1 3 5 7 10 14 18 22 27...
%C Floor of the area under the polygon connecting the lattice points (n, floor(n/2)) from 0..n. - _Wesley Ivan Hurt_, Jun 09 2014
%C Beginning with a(4)=3, the sequence might be called the "off-axis" Ulam-Spiral numbers because they are the numbers in ascending order on the horizontal and vertical spokes (heading outward) starting with the first turning points on the spiral (i.e., 3, 5, 7 and 10). That is, starting with: 3 (upward); 5 (leftward); 7 (downward) and 10 (rightward). These are A033991 (starting at a(1)), A007742 (starting at a(1)), A033954 (starting at a(1)) and A001107 (starting at a(2)), respectively. These quadri-sections are summarized in the formulas of Sep 26 2015. - _Bob Selcoe_, Oct 05 2015
%C Conjecture: For n = 2, a(n) is the greatest k such that A123663(k) < A000217(n - 2). - _Peter Kagey_, Nov 18 2016
%C a(n) is also the matching number of the n-triangular graph, (n-1)-triangular honeycomb queen graph, (n-1)-triangular honeycomb bishop graphs, and (for n > 7) (n-1)-triangular honeycomb obtuse knight graphs. - _Eric W. Weisstein_, Jun 02 2017 and Apr 03 2018
%C After 0, 0, 0, add 1, then add 2 three times, then add 3, then add 4 three times, then add 5, etc.; i.e., first differences are A004524 = (0, 0, 0, 1, 2, 2, 2, 3, 4, 4, 4, 5, ...). - _M. F. Hasler_, May 09 2018
%C Let s(0) = s(1) = 1, s(-1) = s(2) = x, and s(n+2)*s(n-2) = s(n+1)*s(n-1) + s(n)^2 for all n in Z. Then s(n) = p(n) / x^e(n) is a Laurent polynomial in x with p(n) a polynomial with nonnegative integer coefficients of degree a(n) for all n in Z. If x = 1, then s(n) = p(n) = A006720(n+1). - _Michael Somos_, Mar 22 2023
%H Vincenzo Librandi, <a href="/A011848/b011848.txt">Table of n, a(n) for n = 0..1000</a>
%H Kyu-Hwan Lee and Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MatchingNumber.html">Matching Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,4,-3,1).
%F G.f.: x^3*(1-x^2)/((1-x)^3*(1-x^4)).
%F G.f.: x^3/((1+x^2)*(1-x)^3). - _Jon Perry_, Mar 31 2004
%F a(n) = +3*a(n-1) -4*a(n-2) +4*a(n-3) -3*a(n-4) +a(n-5). - _R. J. Mathar_, Apr 15 2010
%F a(n) = floor((n/(1+e^(1/n)))^2). - _Richard R. Forberg_, Jun 19 2013
%F a(n) = floor(n*(n-1)/4). - _T. D. Noe_, Jun 20 2013
%F a(n) = (1/4) * ( n^2 - n - 1 + (-1)^floor(n/2) ). - _Ralf Stephan_, Aug 11 2013
%F a(n) = A054925(n) - A133872(n+2). - _Wesley Ivan Hurt_, Jun 09 2014
%F a(4*n) = A033991(n). a(4*n+1) = A007742(n). a(4*n+2) = A033954(n). a(4*n+3) = A001107(n+1). - _Bob Selcoe_, Sep 26 2015
%F E.g.f.: (sin(x) + cos(x) + (x^2 - 1)*exp(x))/4. - _Ilya Gutkovskiy_, Nov 18 2016
%F A054925(n) = a(-n). A035608(n) = a(2*n+1). _Wesley Ivan Hurt_, Jun 09 2014
%F A156859(n) = a(2*n+2). - _Michael Somos_, Nov 18 2016
%F Euler transform of length 4 sequence [ 3, -1, 0, 1]. - _Michael Somos_, Nov 18 2016
%F From _Amiram Eldar_, Mar 18 2022: (Start)
%F Sum_{n>=3} 1/a(n) = 40/9 - 2*Pi/3.
%F Sum_{n>=3} (-1)^(n+1)/a(n) = 32/9 - 4*log(2). (End)
%F 0 = a(n+2)*(a(n)*(a(n) -6*a(n+1) +4*a(n+2)) +a(n+1)*(8*a(n+1) -10*a(n+2)) + 3*a(n+2)^2) +a(n+3)*(a(n)*(+a(n) -2*a(n+1)) +a(n+2)*(2*a(n+1) -a(n+2))) for all n in Z. - _Michael Somos_, Mar 22 2023
%F 2*a(n) + 2*a(n-2) = (n-1)*(n-2). - _R. J. Mathar_, Feb 12 2024
%e G.f. = x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 10*x^7 + 14*x^8 + 18*x^9 + 22*x^10 + ...
%e p(0) = p(1) = 1, p(2) = 1 + x, p(3) = 1 + x + x^3, p(4) = 1 + 2*x + 2*x^2 + x^3 + x^5. - _Michael Somos_, Mar 22 2023
%p seq(floor(binomial(n,2)/2), n=0..57); # _Zerinvary Lajos_, Jan 12 2009
%t Table[Floor[n (n - 1)/4], {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 28 2011 *)
%t CoefficientList[Series[x^3/((1 + x^2) (1 - x)^3), {x, 0, 70}], x] (* _Vincenzo Librandi_, Jun 21 2013 *)
%t LinearRecurrence[{3, -4, 4, -4, 1}, {0, 0, 1, 3, 5}, {0, 20}] (* _Eric W. Weisstein_, Jun 02 2017 *)
%t Table[Floor[Binomial[n, 2]/2], {n, 0, 20}] (* _Eric W. Weisstein_, Jun 02 2017 *)
%t Table[1/4 (-1 + (-1 + n) n + Cos[n Pi/2] + Sin[n Pi/2]), {n, 0, 20}] (* _Eric W. Weisstein_, Jun 02 2017 *)
%t Floor[Binomial[Range[0, 20], 2]/2] (* _Eric W. Weisstein_, Apr 03 2018 *)
%o (PARI) a(n) = binomial(n, 2)\2;
%o (PARI) vector(100, n, n--; floor(n*(n-1)/4)) \\ _Altug Alkan_, Sep 30 2015
%o (Sage) [floor(binomial(n,2)/2) for n in range(0,58)] # _Zerinvary Lajos_, Dec 01 2009
%o (Magma) [ Floor(n*(n-1)/4) : n in [0..50] ]; // _Wesley Ivan Hurt_, Jun 09 2014
%o (Haskell)
%o a011848 n = if n < 2 then 0 else flip div 2 $ a007318 n 2
%o -- _Reinhard Zumkeller_, Mar 04 2015
%o (GAP) List([0..60],n->Int(Binomial(n,2)/2)); # _Muniru A Asiru_, Apr 05 2018
%o (Python)
%o def a(n): return n*(n-1)//4 # _Christoph B. Kassir_, Oct 07 2022
%Y A column of triangle A011857.
%Y First differences are in A004524.
%Y Cf. A007318, A033991, A007742, A033954, A001107, A006720, A035608 (bisection), A156859 (bisection).
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_, Dec 11 1996
|
